I'm not a coffee drinker, thus have no experience on why slow-drip would be better. Someone please explain the benefits?
Is it just a deeper more thorough infusion of flavor? If I'm ever in Tokyo again maybe It's worth trekking out to Sangenjaya to try it.
The advantage over most other coffee brewing methods is that it gives complete control over infusion temperature, time, and coffee:water ratio. Some other brewing methods offer similar control but vacuum pot is harder to handle and french press leaves a lot of sediment in the cup so pour over (or slow drip as it's being called here) is popular among people who want to bring out the best from their coffees. Like Tead Off pointed out, any cafe that bothers with this method is going to be using high quality lots of coffee which has been properly roasted and is fresh.
Also, with only a couple of exceptions automatic drip machines are really bad, they never have the correct water temperature and there's almost no control over any of the brewing variables. Plus people and cafes that use them often ignorantly fill them with over-roasted, stale, and often rancid coffee.
That said I'm not sure what's special about it for tea? We conventionally brew tea in a tea pot or gaiwan by full immersion like a french press, but even by pouring through a spout we get much fewer leaves coming through than the amount of ground coffee sediment that gets through a french press screen. Slow drip tea will avoid any leaf bits that can scare non-tea drinkers and avoids handing a tea pot full of leaves and hot water to the customer to screw up, but I don't see it making as big a difference for tea as it does for coffee. Unless you're used to tea bags that is.